For a one-off implementation of your typesetting needs, you could try
6.22\mbox{\sc{e}-}21
This will work in both text and math mode. If you use the amsmath package, you should type
6.22\text{\sc{e}-}21
as this is slightly more general than the first solution. Specifically, it'll work better if the expression ever occurs in scriptstyle (first-level sub- and superscripts) or scriptscriptstyle (second-level sub- and superscripts) material.
Conversely, if you have a lot of numbers that you wish to typeset in this way, you should implement a solution along the lines provided in @AlfC's answer. Otherwise, sooner or later some inconsistencies in the way you typeset these numbers will creep in.
$6.22\mbox{\textsc{e}}-21$. I'm assuming this number will be typeset on math mode, right? If you don't want a (typographically correct) math minus sign but a simple dash, be sure to include it in the argument of the\mboxcommand. – Mico Sep 7 '12 at 3:39$6.22\mbox{\sc{e}-}21$is definitely the correct answer, it looks exactly how I'd like it to! You should post that so I can checkmark it. – U2EF1 Sep 7 '12 at 5:22